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Question

(B) Let the time $T$ be expressed as $T = k c^x h^y G^z$,where $k$ is a dimensionless constant.The dimensional formulas are:$[T] = [M^0 L^0 T^1]$$[c]

Vedclass · Accepted Answer

$[c^{-5/2} h^{1/2} G^{1/2}]$

Vedclass · Answer

(B) Let the time $T$ be expressed as $T = k c^x h^y G^z$,where $k$ is a dimensionless constant.The dimensional formulas are:$[T] = [M^0 L^0 T^1]$$[c] = [L T^{-1}]$$[h] = [M L^2 T^{-1}]$$[G] = [M^{-1} L^3 T^{-2}]$Substituting these into the equation:$[M^0 L^0 T^1] = [L T^{-1}]^x [M L^2 T^{-1}]^y [M^{-1} L^3 T^{-2}]^z$$[M^0 L^0 T^1] = [M^{y-z} L^{x+2y+3z} T^{-x-y-2z}]$Comparing the powers of $M$,$L$,and $T$ on both sides:$1) y - z = 0 \implies y = z$$2) x + 2y + 3z = 0$$3) -x - y - 2z = 1$Substitute $y = z$ into equations $(2)$ and $(3)$:$x + 5z = 0 \implies x = -5z$$-(-5z) - z - 2z = 1 \implies 5z - 3z = 1 \implies 2z = 1 \implies z = 1/2$Thus,$y = 1/2$ and $x = -5/2$.Therefore,the expression for time is $T = k \sqrt{\frac{h G}{c^5}}$.

If velocity of light $c$,Planck's constant $h$,and gravitational constant $G$ are taken as fundamental quantities,then express time in terms of dimensions of these quantities.

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